Trigonometric form of complex numbers

Let be the point in the complex plane which is join to the complex number .

The position of the point , and thus of the complex number , we can determine by using numbers and where (the distance from the point to the origin) and is an angle which the segment line closing with the positive part of the real axis.

Then

and

is valid.

Substituting in the expression we obtain trigonometric form of the complex number:

If a complex number is given in the algebraic form , then and we determine from equations:

The last equation gives two solutions for an angle . We choose the angle depend on in which quadrant the number is located. The angle is called an argument of the complex number and is denoted as .

Multiplication and division of complex numbers in trigonometric form

If and , , then their product is define as

and their quotient as

Example 1. Write in the trigonometric form complex numbers and if .

Solution:

We need to determine numbers and .

that is or . Since the number is located in the fourth quadrant, it follows that .

The complex number has the trigonometric form:

The complex conjugate numbers have the same modulus, therefore, for , . is located in the first quadrant, so we have:

Finally, the complex number has the following trigonometric form:

Example 2. Write in the trigonometric form the complex number where:

Solution:

The function cosine is negative in the second and third quadrant, and sine is positive in the first and second quadrant. This means that the given complex number is located in the second quadrant.

Now we have:

That is, or .

We know that the complex number is located in the second quadrant, which means that . The radius of the same complex number is equal to .

Now we can write the given complex number in the trigonometric form:

De Moivre’s formula

For every complex number and

is valid.

The formula above is called the De Moivre’s formula.

Example 3. Calculate

Solution:

Let and .

has the following trigonometric form:

because is in the second quadrant. Therefore

Now, by using the De Moivre’s formula, we have:

For we have:

because the number is located in the fourth quadrant. It follows that the trigonometric form of the complex number is:

By using the De Moivre’s formula we have:

The rest is to calculate the quotient:

th root of complex numbers

Every complex number has exactly different values of -th root given by the formula:

where and .

Example 4. Determine .

Solution:

The number has the following trigonometric form:

Now we have, by using the formula for the th root of a complex number for :

For :

For :

For :

For :

For :

For :

As we can observe, obtained complex numbers are arranged symmetrical. They form the vertices of a regular hexagon.

Similar, holds true in general. The formula for the th root of complex numbers has a geometric interpretation. All th roots of complex numbers have the same modulus , which means that they are equally distant from the origin. Therefore, they all lie on the circle with center at the origin and radius .

Furthermore, arguments of every two consecutive th roots are differ for , which means that these th roots are arranged so that they form the vertices of a regular polygon.